Riccati equations for linear Hamiltonian systems without controllability condition

Peter Šepitka

doi:10.3934/dcds.2019074

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2019, Volume 39, Issue 4: 1685-1730. Doi: 10.3934/dcds.2019074

This issue Previous Article Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions Next Article Entropy rigidity and Hilbert volume

Riccati equations for linear Hamiltonian systems without controllability condition

Peter Šepitka

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-61137 Brno, Czech Republic

Received: July 31, 2017

Revised: June 30, 2018

Published: January 2019

This research was supported by the Czech Science Foundation under grant GA16-00611S

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Abstract

In this paper we develop new theory of Riccati matrix differential equations for linear Hamiltonian systems, which do not require any controllability assumption. When the system is nonoscillatory, it is known from our previous work that conjoined bases of the system with eventually the same image form a special structure called a genus. We show that for every such a genus there is an associated Riccati equation. We study the properties of symmetric solutions of these Riccati equations and their connection with conjoined bases of the system. For a given genus, we pay a special attention to distinguished solutions at infinity of the associated Riccati equation and their relationship with the principal solutions at infinity of the system in the considered genus. We show the uniqueness of the distinguished solution at infinity of the Riccati equation corresponding to the minimal genus. This study essentially extends and completes the work of W. T. Reid (1964, 1972), W. A. Coppel (1971), P. Hartman (1964), W. Kratz (1995), and other authors who considered the Riccati equation and its distinguished solution at infinity for invertible conjoined bases, i.e., for the maximal genus in our setting.

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Mathematics Subject Classification: 34C10.

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